CBSE Notes Class 10 Maths Chapter 7 Coordinate Geometry

1.0Introduction to Coordinate Geometry

Coordinate Geometry, also known as Cartesian Geometry, is a study of geometry using coordinates. In this chapter, we focus on the Cartesian coordinate system and distance formula to solve algebraically geometric problems.

2.0CBSE Class 10 Maths Chapter 7 Coordinate Geometry - Revision Notes

What is the Cartesian Coordinate System?

1. It is a two-dimensional Cartesian coordinate system in which the position of any point can be defined with the aid of two perpendicular lines, and these lines are called axes.

• The horizontal axis is named as the x-axis or Abscissa.
• The y-axis is the vertical axis or Ordinate. 

2. If the x-axis and the y-axis intersect at a point, such an intersection is called origin. The origin is denoted by the letter O with the coordinate (0, 0).
3. Any point in the coordinate plane is represented as P(x, y), where x denotes the distance from the origin along the x-axis and y is the distance from the origin along the y-axis.

The Distance Formula 

In maths, the distance formula is used to find the distance between two points on the cartesian system, for example A(x₁,y₁) and B(x₂,y₂). The distance is always positive. The formula to find the distance between these points: 

$AB=\sqrt{{\left(x_2-x_1\right)}^2+{\left(y_2-y_1\right)}^2}$

The distance is two points if any one point is the origin.  $AB=\sqrt{(x{)}^2+(y{)}^2}$

Example 1: Find the distance between P(2, 6) and Q(8, 3) using the distance formula. 

Solution:

$PQ=\sqrt{(8-2{)}^2+(3-6{)}^2}=\sqrt{36+9}=\sqrt{45}=3\sqrt{5}$

Example 2: Find the point on the y-axis which is equidistant from A(2, 4) and B(4, 8). 

Solution: The point is on the y-axis, hence the coordinates will be (0, y)

$\sqrt{(0-2{)}^2+(y-4{)}^2}=\sqrt{(0-4{)}^2+(y-8{)}^2}$

Squaring both sides,

4 + y² - 8y + 16 = 16 + y² - 16y + 64 

-8y + 16y = 64 - 4

8y = 60 

y = 7.5

Note: If the given point is on the x-axis, then the y-coordinate will be 0 and vice-versa. 

Section Formula 

Section formula is used to find the coordinates of a line segment. For example, in the given figure, C(x, y) is the point cutting line segments A(x₁, y₁) and B(x₂, y₂) in the ratio m and n.

$\mathbf{X}=\frac{m\times x_2+n\times x_1}{m+n}$

$\mathrm{y}=\frac{m\times y_2+n\times y_1}{m+n}$

Here, (x, y) are the coordinates of C. 

Midpoint Formula: 

When point C cuts the line segment AB in two equal parts, meaning m = n, then we can use another formula that is:

$\mathrm{X}=\frac{x_2+x_1}{2}$

$\mathrm{y}=\frac{y_2+y_1}{2}$

Example: Find the coordinates of a point R(x, y) which is cutting the line segment P(2, 4) and Q(4, 8) in the ratio of 2 and 4. 

Solution:

$X=\frac{2\times 4+4\times 2}{2+4}=\frac{8+8}{6}=\frac{8}{3}$

$y=\frac{2\times 8+4\times 4}{2+4}=\frac{16+16}{6}=\frac{16}{3}$

$\text{ Coordinates of }\mathrm{P}\left(\frac{8}{3},\frac{16}{3}\right)\text{. }$

Example: Find the ratio of the point Q that is on the y-axis cutting the line segment R(5, 7) and P(9, 3). 

Solution: Let the ratio be K : 1.

The point Q is on the y-axis; hence, Q(0, y)

Coordinates of Q 

$0=\frac{k\times 9+1\times 5}{K+1}$

k9+15 = 0 

K = 9/5

Hence, The ratio is 9 : 5 

3.0Key Features of CBSE Maths Notes for Class 10 Chapter 7

• The notes are well aligned with the latest curriculum suggested by CBSE. 
• Every concept is simplified and explained with simple and easy language, making these notes ideal for self-learning. 
• The step-by-step guide is provided with every concept of coordinate geometry to ensure better understanding.